What is a general solution to the differential equation #dy/dx=x^3/y^2#?

Answer 1

#y=root3((3x^4)/4+C)#

Treat #dy/dx# like a fraction to move the #y# terms to one side of the equation and the #x# terms to the other:
#dy/dx=x^3/y^2" "=>" "y^2dy=x^3dx#

Integrate both sides:

#=>" "inty^2dy=intx^3dx#

Using the typical integration power rule:

#=>" "y^3/3=x^4/4+C#
Solving for #y#:
#=>" "y^3=(3x^4)/4+C#
#=>" "y=root3((3x^4)/4+C)#
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Answer 2

The general solution to the differential equation ( \frac{dy}{dx} = \frac{x^3}{y^2} ) can be found by separating variables and integrating both sides.

  1. Separate variables: ( y^2 , dy = x^3 , dx ).
  2. Integrate both sides: [ \int y^2 , dy = \int x^3 , dx ]

After integration, we have: [ \frac{1}{3}y^3 = \frac{1}{4}x^4 + C ]

Where ( C ) is the constant of integration.

Thus, the general solution to the given differential equation is: [ y^3 = \frac{4}{3}x^4 + C ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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